Solving Problems using Search

Transcription

1 Solving Problems using Search Artificial Allegheny College Janyl Jumadinova September 11, 2018 Janyl Jumadinova Solving Problems using Search September 11, / 35

2 Example: Romania On holiday in Romania; currently in Arad. Flight leaves tomorrow from Bucharest. Formulate goal: be in Bucharest Formulate problem: states: various cities; actions: drive between cities. Find solution: sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest. Janyl Jumadinova Solving Problems using Search September 11, / 35

3 Example: Romania Janyl Jumadinova Solving Problems using Search September 11, / 35

4 Tree search algorithms Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states). Janyl Jumadinova Solving Problems using Search September 11, / 35

5 Tree search example Janyl Jumadinova Solving Problems using Search September 11, / 35

6 Tree search example Janyl Jumadinova Solving Problems using Search September 11, / 35

7 Tree search example Janyl Jumadinova Solving Problems using Search September 11, / 35

8 (Search) Algorithm Evaluation Completeness: Is the algorithm guaranteed to find a solution when there is one? Optimality: Does the strategy find the optimal solution? Time complexity: How long does it take to find a solution? Space complexity: How much memory is needed to perform the search? Janyl Jumadinova Solving Problems using Search September 11, / 35

9 Search Overview Name Complete Optimal Time Space BFS Yes* Yes* O(b d ) O(b d ) DFS No No O(b m ) O(bm) Greedy best-first No No O(b m )* O(b m ) A* Yes* Yes* exp. exp. b is the branching factor/maximum number of successors of any node, d is the depth of the shallowest solution, m is the maximum depth of the tree * indicates that there is a special case where this may not be true Janyl Jumadinova Solving Problems using Search September 11, / 35

10 Example: Romania Can use Tree Search Algorithms (BFS, DFS - Uninformed or Blind Search). Special cases: greedy search, A search (Informed or Heuristic Search). Janyl Jumadinova Solving Problems using Search September 11, / 35

11 Romania with step costs in km Oradea 71 Neamt 87 Zerind Iasi Arad Sibiu 99 Fagaras Vaslui Timisoara Rimnicu Vilcea Lugoj 97 Pitesti Hirsova Mehadia Urziceni Bucharest 120 Dobreta 90 Craiova Eforie Giurgiu Straight line distance to Bucharest Arad 366 Bucharest 0 Craiova 160 Dobreta 242 Eforie 161 Fagaras 178 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 98 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374 Janyl Jumadinova Solving Problems using Search September 11, / 35

12 Greedy search Evaluation function f (n) = h(n) (heuristic) = estimate of cost from n to the closest goal Example: h(n) = straight-line distance from n to Bucharest. Janyl Jumadinova Solving Problems using Search September 11, / 35

13 Greedy search Evaluation function f (n) = h(n) (heuristic) = estimate of cost from n to the closest goal Example: h(n) = straight-line distance from n to Bucharest. Greedy search expands the node that appears to be closest to goal. Janyl Jumadinova Solving Problems using Search September 11, / 35

14 Greedy best-first search Arad 366 Janyl Jumadinova Solving Problems using Search September 11, / 35

15 Greedy best-first search Arad Sibiu Timisoara Zerind Search Example Janyl Jumadinova Solving Problems using Search September 11, / 35

16 Greedy best-first search Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea Janyl Jumadinova Solving Problems using Search September 11, / 35

17 Greedy best-first search Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea Sibiu Bucharest Janyl Jumadinova Solving Problems using Search September 11, / 35

18 Greedy search Can get stuck in loops, e.g. Iasi Fagaras, Iasi Neamt Iasi Neamt Complete in finite space with repeated-state checking. A good heuristic can give dramatic improvement in search cost. Janyl Jumadinova Solving Problems using Search September 11, / 35

19 A* search Idea: avoid expanding paths that are already expensive. Janyl Jumadinova Solving Problems using Search September 11, / 35

20 A* search Idea: avoid expanding paths that are already expensive. Evaluation function f (n) = g(n) + h(n). g(n) = cost so far to reach n. h(n) = estimated cost to goal from n. f (n) = estimated total cost of path through n to goal. Romania with step costs Janyl Jumadinova Solving Problems using Search September 11, / 35

21 A* Search Arad 366=0+366 Janyl Jumadinova Solving Problems using Search September 11, / 35

22 A* Search Arad Sibiu 393= Timisoara Zerind 447= = Janyl Jumadinova Solving Problems using Search September 11, / 35

23 A* Search Arad Sibiu Timisoara Zerind 447= = Arad 646= Fagaras Oradea 671= Rimnicu Vilcea Sibiu Bucharest Craiova Pitesti Sibiu 591= = = = Bucharest Craiova Rimnicu Vilcea 418= = = Janyl Jumadinova Solving Problems using Search September 11, / 35

24 Local Search So far, addressed a single category of problems: observable, deterministic, known environments, solution is a sequence of actions. Janyl Jumadinova Solving Problems using Search September 11, / 35

25 Local Search So far, addressed a single category of problems: observable, deterministic, known environments, solution is a sequence of actions. Now, relax one of these assumptions: evaluate and modify one or more current states. Janyl Jumadinova Solving Problems using Search September 11, / 35

26 Local Search So far, addressed a single category of problems: observable, deterministic, known environments, solution is a sequence of actions. Now, relax one of these assumptions: evaluate and modify one or more current states. Local search algorithms operate using a single current node (rather than multiple paths). generally move only to neighbors of that node. Janyl Jumadinova Solving Problems using Search September 11, / 35

27 Local Search Benefits 1 use very little memory (usually a constant amount) 2 can often find reasonable solutions in large or infinite (continuous) state spaces for which systematic algorithms are unsuitable 3 useful for solving pure optimization problems, where the aim is to find the best state according to an objective function Janyl Jumadinova Solving Problems using Search September 11, / 35

28 Local Search State-space landscape: elevation is the objective/heuristic function, the goal is to find the global maximum. objective function global maximum shoulder local maximum "flat" local maximum current state state space Janyl Jumadinova Solving Problems using Search September 11, / 35

29 Hill-Climbing Or gradient ascent/descent Also known as Like climbing Everest in thick fog with amnesia objective function global maximum shoulder local maximum "flat" local maximum current state state space Janyl Jumadinova Solving Problems using Search September 11, / 35

30 Hill-Climbing A loop that continually moves in the direction of increasing value that is, uphill. It terminates when it reaches a peak where no neighbor has a higher value. Janyl Jumadinova Solving Problems using Search September 11, / 35

31 Hill-Climbing A loop that continually moves in the direction of increasing value that is, uphill. It terminates when it reaches a peak where no neighbor has a higher value. The algorithm does not maintain a search tree, so the data structure for the current node need only record the state and the value of the objective function. Janyl Jumadinova Solving Problems using Search September 11, / 35

32 Hill-Climbing A loop that continually moves in the direction of increasing value that is, uphill. It terminates when it reaches a peak where no neighbor has a higher value. The algorithm does not maintain a search tree, so the data structure for the current node need only record the state and the value of the objective function. Hill climbing does not look ahead beyond the immediate neighbors of the current state. Janyl Jumadinova Solving Problems using Search September 11, / 35

33 Hill-Climbing Janyl Jumadinova Solving Problems using Search September 11, / 35

34 Implementing Hill-Climbing: Environment Setup Download the program from the class materials repository on GitHub. It has an implementation of the background. It also has definitions for the setup-patches and setup-turtles procedures: Janyl Jumadinova Solving Problems using Search September 11, / 35

35 Next, we need to define and display the highest and lowest points in our terrain: globals [highest ;; the highest patch elevation lowest] ;; the lowest patch elevation - Let s set up two monitors in the Interface tab with the Toolbar (highest and lowest) Janyl Jumadinova Solving Problems using Search September 11, / 35

36 Modify the setup-patches procedure: to setup-patches ask patches [ set elevation (random 10000) ] diffuse elevation 1 ask patches [ set pcolor scale-color green elevation ] set highest max [elevation] of patches set lowest min [elevation] of patches ask patches [ if (elevation > (highest - 100)) [set pcolor white] if (elevation < (lowest + 100)) [set pcolor black] ] end Janyl Jumadinova Solving Problems using Search September 11, / 35

37 Hill-climbing 1 The turtles cannot see ahead farther than just one patch 2 Each turtle can move only one square each turn 3 Turtles are blissfully ignorant of each other ;; each turtle goes to the highest elev-n in a radius of 1 to move-to-local-max ask turtles [ uphill elevation if ( [elevation] of patch-ahead 1 > elevation ) [ fd 1 ] ] end Janyl Jumadinova Solving Problems using Search September 11, / 35

38 Hill-climbing Every patch picked a random elevation, and then we diffused these values one time This doesn t provide a continuous spread of elevation across the graphics window So, we diffuse more! repeat 5 [ diffuse elevation 1 ] Janyl Jumadinova Solving Problems using Search September 11, / 35

39 Hill-climbing Let s plot the number of turtles who have reached the peak-zone (within 1% of the highest elevation) to do-plots set-current-plot "Turtles at Peaks" plot count turtles with [ elevation >= (highest - 100) ] end - Create a slider for the number of turtles and replace the hard-coded value with it Janyl Jumadinova Solving Problems using Search September 11, / 35

40 Hill-climbing We may want to stop the model after all the turtles have found their local maxima - Add turtles-moved? variable to global variables - At the end of the go procedure, add a test to see if any turtles have moved to go set turtles-moved? false move-to-local-max do-plots if (not turtles-moved?) [ stop ] end - In move-to-local-max if a turtle moves, set turtles-moved? to True Janyl Jumadinova Solving Problems using Search September 11, / 35

41 Other Search Algorithms Category Name Uninformed Search Uniform-Cost Uninformed Search Depth-limited Uninformed Search Iterative Deepening DFS Uninformed Search Bidirectional Informed Search Iterative-deepening A* Informed Search Recursive best-first search Informed Search Simplified memory-bounded A* Local Search Simulated annealing Local Search Local beam search Local Search Genetic algorithm Janyl Jumadinova Solving Problems using Search September 11, / 35